#267994

order	512
nilpotency class	3
isoclinism	isoclinic class of #267947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9,5,7)(4,12)(6,10)(8,14)(11,16,13,15)(17,18,23,31,22,20,21,25)(19,24,28,26,29,30,27,32);(1,3,6,5)(2,7,10,9)(4,11,14,13)(8,15,12,16)(17,19)(18,24)(20,30)(21,27)(22,29)(23,28)(25,32)(26,31);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,22,18)(19,26,29,32)(21,25,23,31)(24,27,30,28);(1,5,6,3)(2,9,10,7)(4,13,14,11)(8,16,12,15)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=i,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175190);